內噪音對Brusselator模型穩定態的影響

The effect of internal stochasticity in the route towards the Hopf bifurcation point

謝慧楨(Hui-Zhen Xie)

指導教授 ： 黃敏章

中原大學/理學院/物理研究所/碩士(2016年)

https://doi.org/10.6840/cycu201600538

摘要

確定性系統的穩定態有著不同的形態。由固定點的型態到極限環(limit cycle)的型態之間的區隔稱作霍普分歧(Hopf bifurcation)。另一方面，內在隨機性是化學反應必有的性質。我們利用Brusselator模型，探討由確定性的固定點逼近霍普分歧點時，內在隨機性在功率譜上的表現。我們亦比較了白噪音與色噪音的內在隨機性在影響上的差異。從結果知道了越靠近霍普分歧點時，會使系統的準週期運動(quasi-periodic motion)範圍擴大。色噪音的相關時間大小也確實會影響隨機性，相關時間越大隨機性越小。

關鍵字

內在隨機性, 內噪音, 霍普分歧

並列摘要

There are different types of steady states for a deterministic. The Hopf bifurcation separates the fixed point from a limit cycle. In this work, we intend to give an analysis for the effect of intrinsic stochasticity on a deterministic fixed point in the route towards the Hof bifurcation based on the study of the Brusselator model. The effect is analyzed by studying the power spectrum. Our results indicate that the system first executes a quasi-periodic motion around the vicinity of the fixed point in the phase plane, then the region is enlarged when the Hopf bifurcation line is approached.

並列關鍵字

Internal stochasticity, Hopf bifurcation

參考文獻

[2] J.J. Tyson, The Belousov-Zhabotinskii Reaction, Lecture Notes in Biomathematics 10, Springer-Verlag, Berlin-Heidelberg-New York, 1976.

[5] J. Hizanidis, The Master Equation, 2002.

[6] U. Seifert, Rep. Prog. Phy., 75, 126001, 2012.

[8] J.D. Meiss, Differential Dynamical Systems, SIAM, Philadelphia, 2007.

[9] S. Lynch, Dynamical Systems with Applications Using Matlab, Birkhauser, Boston, 2004.